Explaining the Cross-Multiplication Algorithm

Many high-school mathematics teachers have likely been asked by a student, "Why does the cross-multiplication algorithm work?" It is a commonly used algorithm when dealing with proportion problems, conversion of units, or fractional linear equations. For most teachers, the explanation usually involves the idea of finding a common denominator--one in particular. One might even draw arrows in the first equation, showing how the "a" and the "d" come together, and so on. In this article, the author offers a surprising visual "proof" for the algorithm in a manner that is appropriate for a high-school mathematics classroom. The author also explores some important mathematical ideas appropriate for high-school students, especially if they have been exploring the topic of similarity among geometric figures. (Contains 5 figures.)